Statistics for Business and Economics 7th Edition Chapter Continuous Random Variables and Probability Distributions Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-1 Chapter Goals After completing this chapter, you should be able to:  Explain the difference between a discrete and a continuous random variable  Describe the characteristics of the uniform and normal distributions  Translate normal distribution problems into standardized normal distribution problems  Find probabilities using a normal distribution table Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-2 Chapter Goals (continued) After completing this chapter, you should be able to:  Evaluate the normality assumption  Use the normal approximation to the binomial distribution  Recognize when to apply the exponential distribution  Explain jointly distributed variables and linear combinations of random variables Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-3 Probability Distributions Probability Distributions Ch Discrete Probability Distributions Continuous Probability Distributions Binomial Uniform Hypergeometric Normal Poisson Exponential Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch Ch 5-4 5.1 Continuous Probability Distributions  A continuous random variable is a variable that can assume any value in an interval      thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on the ability to measure accurately Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-5 Cumulative Distribution Function  The cumulative distribution function, F(x), for a continuous random variable X expresses the probability that X does not exceed the value of x F(x) P(X x)  Let a and b be two possible values of X, with a < b The probability that X lies between a and b is P(a  X  b) F(b)  F(a) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-6 Probability Density Function The probability density function, f(x), of random variable X has the following properties: f(x) > for all values of x The area under the probability density function f(x) over all values of the random variable X is equal to 1.0 The probability that X lies between two values is the area under the density function graph between the two values Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-7 Probability Density Function (continued) The probability density function, f(x), of random variable X has the following properties: The cumulative density function F(x0) is the area under the probability density function f(x) from the minimum x value up to x0 x0 f(x )  f(x)dx xm where xm is the minimum value of the random variable x Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-8 Probability as an Area Shaded area under the curve is the probability that X is between a and b f(x) P (a ≤ x ≤ b) = P (a < x < b) (Note that the probability of any individual value is zero) a Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall b x Ch 5-9 The Uniform Distribution Probability Distributions Continuous Probability Distributions Uniform Normal Exponential Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-10 Joint Cumulative Distribution Functions (continued)  The cumulative distribution functions F(x1), F(x2), ,F(xk) of the individual random variables are called their marginal distribution functions  The random variables are independent if and only if F(x1, x , , x k ) F(x1 )F(x )F(x k ) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-59 Covariance  Let X and Y be continuous random variables, with means μx and μy  The expected value of (X - μx)(Y - μy) is called the covariance between X and Y Cov(X, Y) E[(X  μx )(Y  μy )]  An alternative but equivalent expression is Cov(X, Y) E(XY)  μxμy  If the random variables X and Y are independent, then the covariance between them is However, the converse is not true Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-60 Correlation  Let X and Y be jointly distributed random variables  The correlation between X and Y is Cov(X, Y) ρ Corr(X, Y)  σ Xσ Y Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-61 Sums of Random Variables Let X1, X2, Xk be k random variables with means μ1, μ2, μk and variances σ12, σ22, ., σk2 Then:  The mean of their sum is the sum of their means E(X1  X2    Xk ) μ1  μ2    μk Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-62 Sums of Random Variables (continued) Let X1, X2, Xk be k random variables with means μ1, μ2, μk and variances σ12, σ22, ., σk2 Then:  If the covariance between every pair of these random variables is 0, then the variance of their sum is the sum of their variances Var(X  X    Xk ) σ12  σ 22    σ k2  However, if the covariances between pairs of random variables are not 0, the variance of their sum is K K Var(X1  X    Xk ) σ12  σ 22    σ k2  2  Cov(Xi , X j ) i1 j i1 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-63 Differences Between Two Random Variables For two random variables, X and Y  The mean of their difference is the difference of their means; that is E(X  Y) μX  μY  If the covariance between X and Y is 0, then the variance of their difference is Var(X  Y) σ 2X  σ 2Y  If the covariance between X and Y is not 0, then the variance of their difference is Var(X  Y) σ 2X  σ 2Y  2Cov(X,Y) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-64 Linear Combinations of Random Variables  A linear combination of two random variables, X and Y, (where a and b are constants) is W aX  bY  The mean of W is μW E[W] E[aX  bY] aμX  bμY Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-65 Linear Combinations of Random Variables (continued)  The variance of W is σ 2W a 2σ 2X  b 2σ 2Y  2abCov(X, Y)  Or using the correlation, σ 2W a 2σ 2X  b 2σ 2Y  2abCorr(X,Y)σ Xσ Y  If both X and Y are joint normally distributed random variables then the linear combination, W, is also normally distributed Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-66 Example   Two tasks must be performed by the same worker  X = minutes to complete task 1; μ = 20, σ = x x  Y = minutes to complete task 2; μy = 20, σy =  X and Y are normally distributed and independent What is the mean and standard deviation of the time to complete both tasks? Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-67 Example (continued)   X = minutes to complete task 1; μx = 20, σx =  Y = minutes to complete task 2; μy = 30, σy = What are the mean and standard deviation for the time to complete both tasks? W X  Y μW μX  μY 20  30 50  Since X and Y are independent, Cov(X,Y) = 0, so σ 2W σ 2X  σ 2Y  2Cov(X, Y) (5)2  (8)2 89  The standard deviation is σ W  89 9.434 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-68 Portfolio Analysis  A financial portfolio can be viewed as a linear combination of separate financial instruments  Proportion of   Proportion of       Stock   Return on   Stock 1     portfolio value      portfolio value     portfolio   in stock1   return   in stock2   return       Proportion of     Stock N      portfolio value    in stock N   return    Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-69 Portfolio Analysis Example   Consider two stocks, A and B  The price of Stock A is normally distributed with mean 12 and variance  The price of Stock B is normally distributed with mean 20 and variance 16  The stock prices have a positive correlation, ρAB = 50 Suppose you own  10 shares of Stock A  30 shares of Stock B Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-70 Portfolio Analysis Example (continued)  The mean and variance of this stock portfolio are: (Let W denote the distribution of portfolio value) μW 10μA  20μB (10)(12)  (30)(20) 720 σ 2W 10 σ 2A  30 σ B2  (2)(10)(30)Corr(A,B)σ A σ B 10 (4)2  302 (16)2  (2)(10)(30)(.50)(4)(16)  251,200 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-71 Portfolio Analysis Example (continued)  What is the probability that your portfolio value is less than $500? μW 720 σ W  251,200 501.20  500  720  0.44 The Z value for 500 is Z  501.20  P(Z < -0.44) = 0.3300  So the probability is 0.33 that your portfolio value is less than $500 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-72 Chapter Summary  Defined continuous random variables  Presented key continuous probability distributions and their properties  uniform, normal, exponential  Found probabilities using formulas and tables  Interpreted normal probability plots  Examined when to apply different distributions  Applied the normal approximation to the binomial distribution  Reviewed properties of jointly distributed continuous random variables Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 5-73 ... The Uniform Distribution  The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable f(x) Total area under the uniform... The formula for the normal probability density function is  (x  μ)2 /2σ f(x)  e 2π Where e = the mathematical constant approximated by 2.71828 π = the mathematical constant approximated by. .. be transformed into the standardized normal distribution (Z), with mean and variance f(Z) Z ~ N(0 ,1)  Z Need to transform X units into Z units by subtracting the mean of X and dividing by its